The phase of the Riemann zeta function

We, offer an alternative interpretation of the Riemann zeta function zeta(s) as a scattering amplitude and its nontrivial zeros as the resonances in the scattering amplitude. We also look at several different facets of the phase of the zeta function. For example, we show that the smooth part of the zeta function along the line of the zeros is related to the quantum density of states of an inverted oscillator. On the other hand, for Rs > 1/2, we show that the memory of the zeros fades only gradually through a Lorentzian smoothing of the delta functions. The corresponding trace formula for Rs much greater than 1 is shown to be of the same form as generated by a one-dimensional harmonic oscillator in one direction along with an inverted oscillator in the transverse direction. Quite remarkably for this simple model, the Gutzwiller trace formula can be obtained analytically and is found to agree with the quantum result.